Discriminating quantum states: the multiple Chernoff distance

Abstract

We consider the problem of testing multiple quantum hypotheses $\{{\rho }_1^{\otimes n},\dots ,{\rho }_r^{\otimes n}\}$, where an arbitrary prior distribution is given and each of the $r$ hypotheses is $n$ copies of a quantum state. It is known that the average error probability $P_e$ decays exponentially to zero, that is, $P_e=\exp \{-\xi n+o(n)\}$. However, this error exponent $\xi$ is generally unknown, except for the case that $r=2$. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szkoła's conjecture that $\xi ={\min }_{i\ne j}C({\rho }_i,{\rho }_j)$. The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C({\rho }_i,{\rho }_j):={\max }_{0\le s\le 1}\{-\log \mathrm{Tr}{\rho }_i^s{\rho }_j^{1-s}\}$ has been previously identified as the optimal error exponent for testing two hypotheses, ${\rho }_i^{\otimes n}$ versus ${\rho }_j^{\otimes n}$. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szkoła's lower bound. Specialized to the case $r=2$, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.

arXiv commentv2: minor changes

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